Levy-Lees
The Levy-Lees inverse transform maps the Falkner-Skan similarity coordinate \(\eta\) back to the physical wall-normal coordinate \(y\).
The Levy-Lees coordinate used in the Falkner-Skan derivation is
\[
\eta = \frac{u_e}{\sqrt{2\xi}}\int_0^y \rho\,dy'
\]
with
\[
\xi = \int_0^x \rho_e \mu_e u_e\,dx'
\]
For a power-law edge velocity \(u_e = Cx^m\), and with \(\rho_e\mu_e\) constant over the local similarity station,
\[
\xi = \rho_e\mu_e\int_0^x Cx'^m\,dx'
\]
\[
\xi = \rho_e\mu_e C\frac{x^{m+1}}{m+1}
\]
Using \(u_e = Cx^m\),
\[
\xi = \frac{\rho_e\mu_e u_e x}{m+1}
\]
Differentiate the Levy-Lees coordinate with respect to \(y\) at fixed \(x\):
\[
\frac{\partial \eta}{\partial y}
= \frac{u_e}{\sqrt{2\xi}}\rho
\]
Substitute \(\rho = \rho_e/\tau\):
\[
\frac{\partial \eta}{\partial y}
= \frac{\rho_e u_e}{\tau\sqrt{2\xi}}
\]
Now substitute the power-law expression for \(\xi\):
\[
\frac{\partial \eta}{\partial y}
= \frac{1}{\tau}\frac{\rho_e u_e}{\sqrt{2\rho_e\mu_e u_e x/(m+1)}}
\]
Collecting terms gives
\[
\frac{\partial \eta}{\partial y}
= \frac{1}{\tau}\sqrt{\frac{(m+1)\rho_e u_e}{2\mu_e x}}
\]
Therefore the Levy-Lees inverse scale is
\[
\eta_{s,LL} = \sqrt{\frac{(m+1)\rho_e u_e}{2\mu_e x}}
\]
and the physical coordinate is
\[
y(\eta) = \frac{1}{\eta_{s,LL}}\int_0^\eta \tau(\hat{\eta})\,d\hat{\eta}
\]